Higher Chow groups and not necessarily admissible cycles

Abstract

We construct some analog of cubical Bloch's higher Chow groups. Instead of considering cycles in $X\times\mathbb A^n$ we consider varieties $Y$ over $X$ together with a distinguished element in the $n$-th exterior power of the multiplicative group of the field of fraction on $Y$. This definition allows us to make sense of a cycle in $X\times\mathbb A^n$ intersecting faces improperly as an element in this complex. We prove that this complex is well-defined and study its basic properties: flat pullback, the localization sequence etc. As an application we prove that the cohomology of this complex in degrees $m-1, m$ and weight $m$ isomorphic to the cohomology of polylogarithmic complex.